
here \(l\)  and \(m\)  respectively represent the spheroidal harmonics index and azimuthal harmonics index of the mode. \(R_{lm}(r)\)  is the radial wave function and \(S_{lm}(\theta )\)  is the angular wave function.
\(S_{lm}(\theta )\)  satisfies the spheroidal harmonic equation[1]}, [2]}, [3]}, [4]}, [5]}:
\(\frac{1}{\sin \theta }\frac{d}{d\theta }\left(\sin \theta \frac{d S_{lm}}{d\theta }\right)+\left[K_{lm}+(\mu ^{2}-\omega ^{2})a^{2}\sin ^{2}\theta -\frac{m^{2}}{\sin ^{2}\theta }\right]S_{lm}=0~,\) 
